Well, first i would like to say that some of this theories have been pointed out by others, what im trying to do is recopilate them and add some of my own, and all this theories are just fiction to make things coherent in our fanboy heads, because the sad thruth is that fox, just didnt give a f*ck about that.
So ill list miscontinuities, and the solution i give them:
XAVIER.
The problem is xavier ends in First Class invalid and with hair and in Origins, he can walk but without hair. The only explanation i could find is just as emma fost could make the Russian believe she was having sex with him, while she was in the couch, The real Xavier could be inside the "plane" and the one we saw a kind of "Projection". This might sound as an easy way out but anyone ho saw the series or read the comics know, that many times when Xavier projected himself in other minds he could walk inside his mind and the others, so myabe it is just the same thing here. Know with the hair theres plenty of time for him to loose it between movies.The same argument could work at the moment they go jor little Jean grey.
EMMA FROST.
As someone already said the emma frost in Origins could be the child of Emma frost( First Class) and Riptide. So how could this work? Well lets say Emma and Riptide hook up and have Little Emma and Silverfox, while mystique and azazel have nightcrawler, Magneto meanwhile could meet the mother of quicksilver( i dont know who she is). In that way it would make sense, but the most interesting fact here is that when Wolverine rescues the mutants there is Quicksilver and a guy who can turn into a Tornado, so maybe im imagining too much but maybe we can assume, silverfox, little emma and th e guy who turns into a tornado are riptides and emmas child.
BEAST.
Well is a pretty obvious answer the image inducers and its change of fur appearance as in the comics could be an evolution to its mutation.
STRYKER. ( some people dont know this)
The one who appears in First Class is W. Strykers dad, this is no really a theory but a fact but somepeople dont know this.
CYCLOPS.
This is pretty much a confirmation that havok is scotts older brother in the movies because cyclops has a small cameo in Fist Class( pay attention when charles is in cerebro, a children with glasses) and in there his younger so it fits perfectly although it doesnt fit the comics.
Right know i can not remember the other continuity issues some fanboys argued but put them on the comments section and ill try to answer to them in the best way. Im Latin so thank you for having patience with my wirting.
I hope this theories were of your liking. :D